Start by opening your Inbox menu. Tap Edit and select the conversation(s), alert(s), or notification(s) you wish to delete. When you have made your selection, tap Delete.
There is no confirmation step, so make sure that you have made the right selection, as deleted conversations can not be restored by the user that deleted them.
When you delete a conversation, you only delete it from your device. To fully delete a conversation so that it cannot be restored by any user, all other users involved need to delete the conversation.
You can also open a number of notifications and alerts and Confirm seeing them, or tap (X) to dismiss them, depending on the type of notification/alert. This will also remove them from your Inbox.
NOTE: You can start a new conversation with a user after deleting your previous conversation by sending them a new message. However, if a user sends a message to you using a conversation box that you deleted, the conversation will be restored on your device.